Animated Solution for Physics - Laws of Motion: A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate $\frac{dM(t)}{dt}=b{v}^2(t)$, where $v(t)$ is its instantaneous velocity. The instantaneous acceleration of the satellite is

Imagine you are piloting a spaceship coasting through the deep, silent void of space. Your engines are off, and you are gliding at a steady velocity. But space isn't perfectly empty. It's filled with microscopic interplanetary dust. As your ship plows through this dust, it acts like a cosmic vacuum cleaner, sweeping up particles.

Because the dust sticks to your ship, your ship's mass is continuously increasing. This is a classic variable mass system. The question tells us that the rate at which the mass increases is given by:

Where $b$ is a constant and $v$ is your instantaneous velocity. The faster you go, the more dust you sweep up per second! But how does this affect your motion?

Most of us are comfortable with $F=ma$. But here is a catch: $F=ma$ is actually a special case! It only works when the mass of the object is constant. Since our spaceship is gaining mass, we must return to the fundamental, true form of Newton's Second Law, which states that force is the rate of change of momentum ($p$):

Since momentum is mass times velocity ($p=Mv$), we can write:

Because both mass ($M$) and velocity ($v$) are changing with time, we must use the product rule of differentiation to expand this derivative:

Notice that the term $\frac{dv}{dt}$ is exactly what we are looking for—the instantaneous acceleration ($a$) of the spaceship.

Now, let's look at the forces. The spaceship is just coasting; there are no engines firing and no planets pulling on it. Therefore, the net external force is zero ($F_{ext}=0$).

We are given that $\frac{dM}{dt}=b{v}^2$. Let's substitute this into our equation:

Now, it is just a matter of simple algebra to isolate the acceleration. We move the $b{v}^3$ term to the other side:

And divide by the mass $M$:

Look closely at that negative sign. It is incredibly important! It tells us that the acceleration is in the opposite direction of the velocity. Even though there is no external friction in the vacuum of space, the act of sweeping up stationary dust creates an effective drag force. The spaceship has to share its momentum with the stationary dust it collects, which inevitably slows it down.

This beautiful piece of physics shows how mass accretion acts as a natural braking mechanism in the cosmos!